There are several well known mathematical modeling techniques for estimating the risk of a portfolio of financial assets such as securities. The current state of the art involves computing a factor risk model for a universe of potential investments. Factor risk models estimate the risk of thousands or tens of thousands or more of assets using a much smaller set of “factors” that are known to be predictive of the asset returns and volatility. See for example, R. C. Grinold, and R. N. Kahn, Active Portfolio Management: A Quantitative Approach for Providing Superior Returns and Controlling Risk, Second Edition, McGraw-Hill, New York, 2000, and R. Litterman, Modern Investment Management: An Equilibrium Approach, John Wiley and Sons, Inc., Hoboken, N.J., 2003 (Litterman), both of which are incorporated by reference herein in their entirety. These two references present the mathematical details involved in computing and specifying a complete factor risk model including a matrix of factor exposures, a matrix of factor-factor covariances, and a matrix or vector of specific covariances.
Factor risk models are commonly used by investment professionals in a variety of ways. One use of factor risk models is to use them in conjunction with an optimizer to rebalance an investment portfolio. The optimization problem solved would typically use the risk or volatility estimates of the factor risk model to limit the risk of the rebalanced portfolio or part of the rebalanced portfolio. Alternatively, the optimization problem may restrict the exposure of the rebalanced portfolio to one or more factors in the factor risk model. The uses of factor risk models with optimization tools and their potential difficulties is addressed in U.S. Pat. No. 7,698,202.
Another use of factor risk models is for ex-post performance analysis of a history of investment portfolios. Such performance attribution can be performed in a number of different well known methods. For example, the history of predicted risk and factor exposures as defined by a history of factor risk models and a history of investment portfolios is often part of a factor-based performance attribution. In this kind of analysis, the history of factor returns is important, as it is used to explain the sources of return and risk in the history of investment portfolios. Groups of factors in a factor risk model such as industries or sectors can also be used for a Brinson-style or returns-style attribution. Details involved in the expost performance analysis are given in Litterman.
Some factor risk models are constructed for a universe of potential investments that all trade at the same time. For example, most US equities trade from 9:30 AM to 4:00 PM Eastern Standard Time. A factor risk model for US equities can use the prices as of the close of the US market and avoid most issues related to asynchronous prices and returns.
However, American Depository Receipts (ADR) are tradable, equity assets that are traded on the US market but replicate a portfolio of non-US stocks. For example, a Japanese ADR trades on a US exchange but tracks the performance of a portfolio of Japanese stocks. Since the US and Japanese markets do not trade at the same time, it is possible that the asynchronous closing prices of the ADR and the underlying Japanese equities may distort the relationship between their returns.
In addition, many factor risk models cover assets that trade in more than one market. For example, a global factor risk model includes assets that are traded around the world. Such factor risk models have a need to address the potential distortion caused by asynchronously traded assets. Addressing this potential distortion would improve the performance of the factor risk model.
The issue of asynchronous prices and their potential distortion of financial computations has been studied previously.
The issue of autocorrelation between daily prices and their distortion of the estimate of the covariance of asset returns was discussed in W. K. Newey and K. D. West, “A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix,” Econometrica, 55(3), pp. 703-708, 1987, and W. K. Newey and K. D. West, “Automatic Lag Selection in Covariance Matrix Estimation,” The Review of Economic Studies, 61(4), pp. 631-653, 1994, both of which are incorporated by reference herein in their entirety. In these two papers, the authors demonstrate that market microstructure and market timing can create serial dependence in daily asset returns. For example, lead-lag relationships often occur in which an assets price is more likely to go up on a day following a price increase than on a day following a price decrease.
Newey and West propose a correction algorithm that can be applied to either a factor risk model's factor-factor covariance matrix or an asset-asset covariance matrix. This correction algorithm compensates for autocorrelation of either the factor returns or the asset returns. This correction algorithm applies to any set of covariance data which exhibits a true, bona fide autocorrelation; that is, any set of returns for which a genuine lead-lag relationship across assets or factors exists regardless of the sampling of the data. Such an autocorrelation can exist between assets and factors of a single trading market, such as the US equities market. The Newey-West algorithm does not reduce spurious autocorrelation created by returns from assets or underlying assets on markets that trade at different times. Such spurious autocorrelation is purely a result of data-sampling. In other words, the Newey West correction implicitly assumes (in the context of multi-factor risk models) that the asset returns are already aligned and corrects for their natural autocorrelation behavior. Thus, there is a need to create a method for correctly estimating the covariance of assets and factors whose asset prices and returns are asynchronously timed. Such a model will improve the covariances and correlations in any factor risk model. It is also likely to reduce the observed autocorrelation of factor returns in factor risk models. A good asynchronous timing model of asset prices and returns is likely to improve the Newey-West correction normally applied to factor risk models.
Scholes and Williams describe a modified estimation procedure for betas from asynchronous data in Scholes, M., and J. Williams, “Estimating Betas From Nonsynchronous Data”, Journal of Financial Economics, 5, pp. 309-328, 1997, which is incorporated by reference herein in its entirety. This paper describes an approach to compensate the estimate for alphas and beta (the ratio of two covariance estimates) for assets that are traded asynchronously. However, the asynchronously studied is driven by the frequency at which assets are traded and is modeled as by a randomly distributed variable. In their model of asynchronous returns, the relative timing of the markets upon which the assets are traded is not considered. In fact, the principal illustrative example used is a comparison of stocks traded on the New York Stock Exchange and the American Stock Exchange, both of which trade during the same hours. By analyzing the manner in which least squares estimates are biased by returns generated by randomly traded assets, this work creates a correction algorithm for betas. However, it does not address any errors caused by known differences in the times of market trading, nor does it indicate how their approach could be utilized when creating a factor risk model.
Burns, Engle, and Mezrich propose using a vector, moving average model to improve the estimates of asset-asset covariance in “Correlations and volatilities of asynchronous data,” by Patrick Burns, Robert Engle, and Joseph Mezrich, University of California at San Diego, April 1998, available at URL http://ideas.repec.org/p/cdl/ucsdec/97-30r.html, which is incorporated by reference herein in its entirety. This study does address the issue of asset returns for assets traded on markets that trade during different hours. However, this work does not show how such models may be easily utilized when estimating factor risk models. It does not discuss which components of a factor risk model—factor returns, types of factors, factor covariance, specific return, and specific risk—are affected by the model of synchronized prices. Nor does it describe efficient computational methods that can be used to incorporate the model into a factor risk model.
The invention disclosed here gives the first practical approach by which a model for correction of asynchronous returns may be simply and efficiently incorporated into the factor regression of a multi-factor risk model. Furthermore, the invention explicitly describes how depository receipts may also be modeled in a multi-factor risk model using a model of asynchronous returns.